Deterministic preparation of highly non-classical macroscopic quantum states

We present a scheme to deterministically prepare non-classical quantum states of a massive mirror including highly non-Gaussian states exhibiting sizeable negativity of the Wigner function. This is achieved by exploiting the non-linear light-matter interaction in an optomechanical cavity by driving the system with optimally designed frequency patterns. Our scheme reveals to be resilient against mechanical and optical damping, as well as mechanical thermal noise and imperfections in the driving scheme. Our proposal thus opens a promising route for table-top experiments to explore and exploit macroscopic quantum phenomena.


I. INTRODUCTION
Non-classicality of mechanical motion has recently been a topic of great interest both theoretically and experimentally as it represents a test ground to address many important questions ranging from quantum-to-classical transitions and collapse models [1][2][3][4][5] to the interface between quantum mechanics and gravity [6,7].While we have extensive literature that has focused on the quantumness of microscopic objects, it is a challenge to deterministically isolate genuine quantum features that can be accessed in experiments, and few experiments with coherent superpositions of quantum objects with large mass exist [8,9].
Massive mechanical oscillators have been intensively investigated in quantum optomechanics [10][11][12][13], and optomechanical cavities are regarded as an optimal framework to make clear comparisons between the predictions of classical theory and their quantum counterparts [14][15][16][17].Indeed, they were proven to exhibit a large degree of macroscopicity, reaching µ = 19 on a scale where the Mach-Zender interference of Cs [18] and the Schrödinger gedanken experiment reach values of µ = 10.6 and µ ∼ 55 respectively [19].
Thanks to their peculiar properties, these systems have been historically studied in the context of force sensing [20,21] and for the preparation of non-classical states of the mechanical motion, such as squeezed states [22][23][24], single phonon excitations [25][26][27] or even Schrödinger cat states [16].However, none of these proposals has come up with the deterministic creation of a macroscopic state, relying instead on a probabilistic approach [28,29] or on a transient regime [30,31].
In this article we show how the non-linear light-matter interaction between an electromagnetic field and a movable mirror in an optomechanical cavity can be exploited in an optimal fashion by driving the cavity with appropriately chosen pump profiles.This enables the deterministic preparation of a range of quantum states of the mirror such as squeezed states and non-Gaussian coherent superpositions exhibiting sizeable negativity of the Wigner function.
Starting from the ideal situation when the system is initially cooled down to the ground state and performs a unitary evolution, we further analyse the resilience of our scheme to mechanical and optical damping, as well as to thermal excitations and imperfections in the driving pattern.
We show that the control scheme developed in this manuscript permits maximally non-classical states to be achieved, which makes it ideal for accurate tests of decoherence models and potential limitations on coherent superpositions of massive objects.
-In Sec.II we introduce the model of an optomechanical cavity and present the Magnus expansion technique to perturbatively solve a non-linear time dependent Schrödinger equation.
-Sec.III contains the main result of our paper, as it is devoted to the computation of the driving profiles that have to be applied to obtain the desired quantum states of the mirror, i.e. squeezed states (III A) and highly nonclassical coherent superpositions (III B).Sec.III C reviews standard measurement techniques to experimentally readout such non-trivial mechanical states.-In Sec.IV, we analyse and characterise the states of the mirror in terms of the negativity of their Wigner function and of a rigorous quantification of quantumness.We also provide evidence of the validity of the utilized perturbative approach, and we devote an extensive subsection (IV C) to discuss the resilience of the control scheme to a range of experimental imperfections, analysing the effects of optical and mechanical decoherence, initial mechanical thermal noise and imperfect optical driving.

A. The Optomechanical setup
We consider an optomechanical cantilever modelled as a harmonic oscillator and interacting with a light field through radiation pressure in the single mode approximation (see Fig. (1)).The free evolution of the system is given by H 0 = ω c a † a + ωb † b, where ω (ω c ) is the mechanical (cavity resonance) frequency and b and b † (a and a † ) are respectively the annihilation and creation operator of the mirror (cavity field).The interaction couples the intensity of the light field with the position of the mechanical element and is described by H int = −ga † a(b+b † ) [32], where g = (ω c /L) /(2mω) is the coupling constant, L is the length of cavity at equilibrium and m is the mass of the mirror.Adding external driving ξ(t) of the cavity, the complete Hamiltonian of the system reads and generally induces correlations between both subsystems.A correlated state, however, implies that a mixed quantum state needs to be attributed to each subsystem alone, or that the measurement on one of the subsystems results in the probabilistic preparation of the other.The goal of the present paper lies in finding the driving pattern ξ(t) such that the cubic optomechanical interaction creates non-trivial states of the mirror without cavity-mirror correlations.In particular, the chosen driving profile will also ensure that the cavity ends up in its initial state, which will significantly ease the readout subsequent to the state preparation.Indeed, most of the current state reconstruction techniques of mechanical motional states are achieved through homodyne tomography of a probe light field, i.e. the so called back-action-evading interaction [24,33,34].It is therefore an essential requirement that the cavity is in its well defined initial state when the read out of the mechanics is performed.

B. The time evolution
In the limit of weak coupling g/ω = k 1, which is in agreement with state-of-the-art experiments operating at k 10 −2 [10,13], we can solve the dynamics in a perturbative expansion in powers of k.To this end, it is helpful to first find the time-evolution operator U 0 (t) induced by the non-interacting Hamiltonian H 0 (t).Since H 0 (t) is harmonic, U 0 (t) can be constructed exactly, even though H 0 (t) is time-dependent.The explicit solution satisfying i U0 (t) = H 0 (t)U 0 (t) and U 0 (0) = 1 reads t1) , and With this at hand, one can extract the interaction Hamiltonian in the frame defined by the harmonic motion as which explicitly reads where X m (t) = b † e iωt + be −iωt .
Because of the cubic nature and the time-dependence, it is not possible to analytically solve the time-evolution operator V (t, t 0 ) induced by H I (t) exactly, but it can be obtained in the perturbative Magnus series [35] where the individual terms (the last integration with d 3 τ = dt 1 dt 2 dt 3 is taken over t ≥ t 1 ≥ t 2 ≥ t 3 ≥ t 0 ) and similar higher order terms satisfy the proportionality M j (t, t 0 ) ∼ k j for any given driving profile ξ(t).
Given the explicit form of H I (t) in Eq. ( 4), the lowest order term M 1 (t) is an interaction that induces correlations between cavity and mirror.The higher order expansions M j (t) (j > 1) will generally contain both interaction and single-particle terms of mirror or cavity alone.Since the central goal of our work is deterministic state preparation, we will require that M 1 (t) and undesired terms in M j (t) (j > 1) vanish at the final instance in time N T after N periods T = 2π/ω of the mechanical motion.In the following, we will show how to design driving profiles ξ(t) such that all interaction terms and all single-particle terms of the cavity vanish at t = N T , but such that the single-particle terms of the mirror induce highly non-classical states.
For the identification of these driving profiles it will prove useful to express the propagator V (T N, 0) over N periods as where it is implied that terms are ordered with decreasing value of s in the product; the M (s) are defined via the relation exp(−iM (s) ) = V (T s, T (s − 1)), and can be expanded in the series analogously to Eq. ( 6).The Baker-Campbell-Hausdorff relation yields and similar relations for higher order terms.For a general driving ξ(t), there is no reason to expect that the M (s) 1 add up to zero, or that undesired terms in the summation for M 2 (N T, 0) cancel each other.We can, however, consider time dependent driving profiles ξ s (t) resulting in different interaction Hamiltonians H with M (s) = W † s M (1) W s .Since all terms now depend on the W k , which can be chosen freely, we will benefit from this freedom to ensure that any undesired term in M j vanishes or is modified as desired.As we will see in the following there are clear physically motivated choices for the W k that achieve the aim, and that translate into rather simple driving profiles.

III. STATE PREPARATION
Due to the large separation of the resonance frequencies of cavity and mirror, it is essential to drive the cavity close to the sidebands with frequencies ω c ± ω to enable the exchange of excitations between the two subsystems.In Sec.III A and III B we will find suitable profiles such that the mirror evolves into a strongly squeezed state and a state with pronounced non-Gaussian and non-classical features respectively.Apart from an interest in its own, the discussion on strongly squeezed states in Sec.III A shall help to exemplify the framework developed above, with simpler algebra than found in the preparation of non-classical states.

A. Preparation of macroscopic squeezed states
Mechanical squeezing is obtained via a bi-chromatic driving with detunings ±ω with respect to the cavity resonance.The corresponding driving profile with amplitude E results in the lowest order contribution to the Magnus expansion with the cavity and mechanical quadrature operators , as well as the unitless amplitude and coupling constant η = E/ω and k = g/ω.This suggests the particularly simple choice W s = (exp(−in c ϕ s )), that rotates cavity operators in phase space as with The required driving profiles can be obtained by reverse-engineering the derivation of the interaction Hamiltonian (see Eq. ( 4)) and are rather elementary to implement.In fact, different driving periods differ from each other merely by the phase shift ϕ s , and simple bi-chromatic driving with a step-like phase variation realises the desired dynamics.With Eqs.(11) and ( 12) one directly obtains such that any undesired interaction terms cancel for any choice satisfying s exp(iϕ s ) = 0.The second order expansion reads FIG. 2. Phase space schematic representation of the algebra that lies under the sum of exponential operators that are rotated by angles ϕ l .As illustrated on the left, we consider the case where the set of phases satisfies N l=1 e iϕ l = 0. On the right we show that the total area spanned by the representation of such phases in phase space is related to the sum over all the combinations of commutators: The choice s ϕ s = 2π directly implies s W † s P c W s = 0, so that the interaction terms m I,s 2 add up to 0 as desired.In addition, given the relation it is desirable to achieve s e i2ϕs = s e −i2ϕs = 0, which eventually motivates the selection ϕ s = 2π(s − 1)/N (assuming N > 2).The remaining terms in m c,s 2 depending on n c are independent of the choice of ϕ s and can not be modified.The last, and most important, contribution to M 2 is given by Eq. (15).With the choice ϕ s = 2π(s − 1)/N , the sum over the sin-terms results in which scales ∼ N 2 /(2π), as illustrated in Fig. 2.
All-together, we have thus arrived at dynamics, such that no results of an interaction appear at the final instance in time and no excitations in the cavity have been created.Up to a global phase factor, which we will henceforth always neglect, the full propagator reads V m describes the dynamics of the mirror and corresponds to a squeezing operation.Indeed, up to a rotation by an amount δ = arctan(|ζ|), it can be recast in the form ) with the squeezing parameter that scales as |ζ| ∼ N 2 .Thanks to this quadratic dependence, one obtains substantial squeezing after a few intervals.Besides, we should keep in mind that the perturbative regime requires reasonably short propagation times, i.e. small values of N , and the present analysis is valid in the limit k 1, as the neglected third order term scales as M 3 ∼ k 3 η 2 N .For a reasonably weak interaction, k = 1/400, and sufficiently strong driving, η = 10, one achieves square displacement of the position with a squeezing of the momentum quadrature resulting after N = 20 periods in ∆P 2 m = 0.5 and ∆X 2 m 8.9.

B. Non-classical Quantum States
Let us now discuss the creation of non-classical states, which requires to suppress not only interaction effects, but also Gaussian contributions to the dynamics, since these will tend to drive the system towards classical states.We will therefore double the detuning as compared to Sec.III A, but employ qualitatively similar driving profiles with phase shifts ϕ s whose form is to be determined.Thanks to the chosen detuning, the first order Magnus term M 1 vanishes irrespectively of the choice for the ϕ s .
The second and third order terms are as well as M (1) and Following Eq. ( 8), the second and third order contribution to the generator of the dynamics over n periods read 3 -in general, there would be contributions resulting from noncommutativity of M  19) and ( 20) display a rather complicated form reflecting the complex dynamics induced by the non-linear Hamiltonian, it is still possible to ensure the desired goals of a product state with an empty cavity and a non-classical state of the mirror.The central property of the interaction terms m I 2 and m I 3 that we need to exploit is that every element in W † s m I j W s (j = 2, 3) is proportional to exp(±iϕ s ) or exp(±i2ϕ s ).That is, similarly to the creation of squeezed states discussed above in Sec.III A, the choice ϕ s = 2π(s − 1)/N ensures that all interaction terms cancel each other.
Making use of all the cancellations, we finally arrive at the desired separable propagator V c does not create any excitations in the cavity, so that the cavity vacuum state is preserved by the dynamics, and the cubic mirror operator Q m induces highly non-classical states, which we will examine in detail in Sec.IV A below.
Prior to an accurate discussion on the accessible states, however, it is worth taking a critical look at the perturbative expansion.Since the relevant term ∼ Q m in Eq.( 20) scales as k 3 η 2 , the cubic dependence on k will make an experimental realisation more challenging than the creation of squeezed states which is a second order effect.Given the quadratic dependence on η 2 , however, strong driving can compensate for the weak interaction.Yet, in the strong driving regime special care needs to be taken in the perturbative expansion: so far we were only concerned with powers of k, but for sufficiently large values of η, a high power of η can make a term relevant despite its high order in k.In particular, M 2 in Eq.( 19) contains terms ∼ (kη) 2 n c , as well as it generates fourth order contributions ∼ (kη) 4 n c resulting from the commutators [M 2 ].This is not a severe issue for the state preparation, since such terms describe neither an interaction between cavity and mirror nor they create excitations in the system.They do induce, however, a back-action of the dynamics, rotating the field operators at each period and spoiling the effect of the previously engineered phase shifts W s .As a result of that, the propagator at the end of the driving time does not factorize into individual propagators of mirror and cavity.It thus becomes necessary to change the driving profile accordingly to compensate for this effect.In order to do so, it is instructive to rewrite the propagator over the first interval, neglecting terms of order k 4 η s with s < 4 and terms of order k s with s > 4 as , with 3 , and mc Recalling Eq. ( 9), the propagator over N periods can be written as and we should choose the W s such that the prefactors W s+1 W † s cancel the term e i 4 3 πk 2 η 2 nc in Eq. ( 22).This is achieved with the phase shift which counterbalances exactly the phase shift ∆ = 4π 3 k 2 η 2 that the cavity experiences through the driving over each period as described in Eq. (22).With this, the propagator reads and the basic principles developed above apply.Quite importantly, however, the terms ∼ (kη) 2 n c no longer appear, and the only remaining contribution scaling as ∼ (kη) 2 in m c 2 in Eq. ( 22) is the polynomial a † 2 + a 2 + 1. Operators a † 2 and a 2 cancel out exactly in the summation defined in Eq. ( 8) thanks to the specific set of phase shifts, and the '+1' brings an irrelevant global phase.The terms ∼ (kη) 4 arising from the commutators [M 2 ] contribute either to the global phase or are proportional to n c (since [a 2 , a † 2 ] = 4n c + 2).Lastly, we will see that the only term ∼ (kη) 4 in M (1) 4 depends on P c and averages out in the summation over the N periods.One thus ends up with the same propagator as in Eq. ( 21) up to a perturbative modification in the cavity phase shift C. Readout The final readout of the mechanical motion is a matter that has already been widely analysed theoretically [33,36] and implemented experimentally [24,34] with high precision.The most promising technique to perform quantum state reconstruction is called back-action-evading interaction and is based on state transfer.When the mirror is in the state of interest and the cavity is empty, a red detuned laser with frequency ω d = ω c − ω induces exchange of excitations from the former to the latter.Hence, tomography of the prepared mechanical state of the mirror can be carried out through homodyne measurement of the light leaking out of the cavity [37,38].Moreover, since we ensure that there are no residual correlations between cavity and mirror when the measurement protocol is applied, the desired mechanical quantum state is deterministically read out.

IV. STATE ANALYSIS
In contrast to the well characterised squeezed states discussed above in Sec.III A, it is not clearly established what type of states are obtained with the cubic generator V (3) m (N ) at the third order in the coupling (see Eq. ( 21)).
Hence, in order to provide further information on the mechanical states achieved with such non-linear dynamics, we construct V After the analysis of the achievable mechanical states, we will devote this section both to the accuracy of the perturbative approach and the resilience to various possible sources of error.

A. Non-classicality
The quantum state ρ of the mirror can be represented in terms of the Wigner function which is a quasi-probability distribution in phase space spanned by momentum and displacement variables p and q.Fig. 3a) depicts the Wigner function for the state |Ψ( 20) Ψ( 20)|.Quantumness can be characterised by oscillations of W (q, p) including negative values.The amplitude of these oscillations is linked to the degree of coherence, while their wave-length is inversely related to the macroscopicity of the state.As one can see, the Wigner function of |Ψ( 20) features high wavelength oscillations with large amplitudes.This is visible on a more quantitative level also in Fig. 3b) which shows the cut W (q, 0) through the Wigner function.
In order quantify the non-classicality, we resort to the estimator of quantumness [39]

B. Perturbative regime
All considerations so far apply to the perturbative regime k 1, and it is essential to gauge the range of applicability of the perturbative approximation.To this end, we compare the dynamics obtained with the Magnus expansion in third-and fourth-order approximation.Analogously to the analysis in Sec.III B for V (3) m (N ), we find the dynamics of the mirror after N mechanical periods at fourth order with ζ (4) = (πk 2 η) 2 N cot(π/N ).Hence, we recall an important figure of merit, i.e. the state fidelity between two quantum states, which is defined from their density operators A and B as This quantity can be used to provide an estimate of the accuracy of the third order Magnus preparation 3 by comparing it with the fourth order state 4 obtained numerically propagating Eq.( 27).Fig. 5 depicts F 3,4 as function of the integration time.We infer that the deviations between 3 and 4 are in the permille regime for the first 10 driving periods, and even at N = 20, the third order approximation is accurate within 1%.We deem an error of 1% below the accuracy of what could be achieved experimentally within the next years, and thus feel that the perturbative treatment is highly adequate for the present purpose.

C. Experimental imperfections
Having laid out the general principles of deterministic state preparation and discussed the set of macroscopic states that are accessible with different drivings, it is appropriate to gauge how unavoidable experimental imperfections will affect the desired process.To this end, we analyse the impact of optical and mechanical decoherence, initial thermal excitations in the mirror and imperfect phase shifts of the driving fields.

Optical decoherence
Due to experimental difficulties, the most delicate aspect affecting the unitarity of the evolution, and consequently the preparation of the desired mechanical state, is attributable to optical losses.The leaking of photons from the cavity is quantified by the decay rate κ, which is defined as the inverse of the time light remains in the cavity.A common approach to include the effects of such photon losses in the dynamics is to express the evolution of the system density matrix ρ in terms of the Master equation ρ where H is the system Hamiltonian and L[ρ] is the Lindblad operator When operating in the so called resolved sideband regime with κ ω, that is realised in many current experiments [13,22,40,41], the impact of photon loss can be captured in terms of a perturbative solution of the Master equation.To this end, we consider the Master equation for ρ = V (t)ρV † (t), where V (t) satisfies i V = HV , which is obtained as detailed in Secs.II B -III.
Solving this perturbatively, yields the contribution of photon loss to the dynamics in terms of powers of κ/ω.The integration for a generic time t assumes a rather complex form and correlations between field and mirror are created because of dissipation.However, thanks to the very specific bi-chromatic driving pattern many terms cancel out at the end of the evolution.Actually, our proposed set of constant phase shifts {ϕ s } is crucial to suppress the majority of these unwanted non-unitary and de-coherent contributions, including all correlation terms proportional to the driving amplitude η.
More specifically, in leading order in k and κ, we obtain at t = NT an expression that is completely independent of η and is thus well suited to describe the strong driving regime This result has a very clear physical interpretation: since the light-matter interaction conditionally displaces the mirror by an amount proportional to the number of photons in the cavity, each photon that has leaked out of the resonator should then be matched with a missing displacement e g(b † −b)/ω of the mirror as indicated in Eq. (30).
Fig. 6 depicts the state fidelity (see Eq.( 28)) after N = 20 mechanical periods as a function of the ratio κ/ω between the full state of the system (cavity plus mirror) obtained in the leaking scenario in Eq.( 29) and the ideal one predicted by Eq.( 21).As one can see, a loss rate satisfying κ/ω < 10 −2 results in a reduction of the fidelity by 3%.This condition, together with the strong driving regime, is in accordance with Ref. [42], where the resolved sideband regime and the condition g/κ > 1 were theoretically derived as requirements to resolve the granularity of the photon stream and fully exploit the non-linearity of the system to observe purely quantum features.

Thermal initial state of the mirror
Since the evolution operator in Eq.( 21) factorises into a propagator for the mirror and a propagator for the cavity, one obtains a product state of mirror and cavity for any initial product state.That is, there is no fundamental need to require the mirror to be initially cooled exactly to the ground state, but initial thermal excitation of the mirror will affect the non-classicality of the final state.Fig. 7 depicts cuts through the Wigner function of the mirror after 20 periods of driving for different initial thermal populations with n th m = 1 and n th m = 10, i.e. above the experimental threshold of n th m ∼ 0.2 achievable with sideband cooling (at a mechanical frequency ω = 2π × 10 7 Hz) [34,41].The strong oscillatory behaviour with negative values of W is clearly displayed for an initial state with n th m = 1.Only for n th m = 10, i.e. substantially above the limits of side-band cooling, the quantum mechanical features are mostly washed out by the thermal contributions.

Mechanical decoherence
The main source of mechanical decoherence for a cooled optomechanical resonator arises from mechanical damping, which is characterised by the damping rate γ m at which a phonon excitation is lost in the environment.This process is conceptually analogous to optical photon losses from the cavity which happen with rate κ.Current experiments have achieved mechanical damping substantially below photon loss (γ m κ), what suggests that mechanical decoherence will not be a limiting factor.Since, however, highly non-classical, coherent superpositions of macroscopically distinct states are particularly sensitive to decoherence, a critical assessment of motional decoherence is in order.To this end, we use a stochastic perturbative model of mechanical damping.We assume coherent dynamics during each period T , after which the state vector suffers a phonon loss described by the jump operator (1 ⊗ b m ) |Ψ with probability p = γ m T .A mixed state of the mirror is obtained by averaging over such processes, and since p is sufficiently small, one can safely restrict the average to processes including at most two phonon losses.Fig. 8 depicts the state fidelity after N = 20 periods of driving as a function of Q.Despite the general sensitivity of non-classical states to decoherence, the impact of mechanical damping on the state fidelity is of order 10 −4 , and thus negligible as compared to the other imperfections discussed above.

Laser driving
The central goal of deterministic state preparation is achieved through the application of appropriate phase shifts after each period of driving.As a last experimental imperfection, we consider the impact of deviations from the ideal driving profiles with the step-like phase shifts ϕ s described in Sec.III.To this end, let us replace the discontinuously evolving phase ϕ(t) =  T is a linearly increasing phase factor and each term ϕ l (t) = A l sin(lωt) oscillates with frequency lω and amplitude A l .The set of amplitudes is chosen such that at any order d, ϕ (d) c (t) is tangent to the step function in the centre of the step, i.e. for t = (2jπ + 1)/ω with j ∈ [0, N − 1] (see Fig. 9 for a graphical representation).
Thanks to the continuous time dependance, it is then possible to analytically compute the generator with a Magnus expansion (as discussed in Sec.II B), and subsequently numerically integrate the dynamics over N mechanical periods.Interestingly, we obtain a separable propagator at every order d, without correlations between mirror and cavity, and which will still result in deterministic state preparation.
Most importantly, while resorting to the sole linear function 2π N t T single-particle terms of the cavity do not completely cancel out, resulting in a final average population n c ∼ 0.2 n m , these contributions are efficiently suppressed already at the order d = 3, when n c ∼ O(10 −7 ) n m (see Fig. 9).This is an essential requirement since cavity excitations could potentially prevent the final readout through back-action-evading inter-  action.Remarkably, the final non-classical mechanical state of the mirror obtained with these imperfect driving pattern presents a very high fidelity F 0.98 with the ideal step-like case.

V. CONCLUSIONS AND OUTLOOK
Our scheme reveals to be resilient to various potential experimental imperfections, and in particular its robustness against mechanical damping supports various applications.For example, the massive mirror can be used as continuous variable quantum memory, as it has already been proposed in Ref. [43], or as probe for decoherence.The quantumness I of the mirror is an extremely sensitive indicator of any type of mechanical decoherence and is thus ideally suited to probe fundamental physics such as gravitationally induced effects on the mechanical motion.
It should also be highlighted that there are no stringent requirements to the initial cooling condition.Here we required n th m 1 such that no large thermal contributions outshine the limited number of excitations induced by the cubic propagator.The utilised approach to find optimal driving patterns is however easily extended to higher orders in the Magnus expansion and correspondingly longer propagation times and/or larger coupling k, which would give rise to more highly excited states and hence to measurable quantum effects also in case of higher initial thermal noise.
The present control scheme is also not necessarily restricted to the mirror-cavity setup discussed here, but similar driving patterns can also be applied to a variety of systems that share similar non-linear hamiltonians such as atomic spin ensembles, trapped atoms or levitated nanoparticles [44].

FIG. 1 .
FIG.1.Conceptual scheme of the archetypical optomechanical setup.A laser field with envelope ξ(t) is used to pump a Fabry-Pérot cavity with resonance frequency ωc.One cavity mirror is movable and modelled as harmonic oscillator with mass m and frequency ω.When the light escapes the cavity it is rotated by a λ/4 wave-plate, reflected by a polarizing beam splitter (PBS) and measured interferometrically with respect to a reference beam.

I
(t) in each interval.With the specific choice H

( 3 )
m (N ) numerically in a truncated Hilbert space including up to 80 × 10 3 excitations.As prototype for discussion, we consider the state |Ψ(20) = V (3) m (20) |0 obtained after N = 20 periods of driving with the mirror initially in its ground state.As specific parameter values we choose η = 20 and k = 1/60 consistently with the perturbative expansion.

FIG. 3
FIG. 3. a) 3D Wigner function of the mirror after 20 driving periods and b) its profile when it is cut by the plane p = 0.The experimental parameters are set as η = 20, k = 1/60 and the resulting average population is b † b 20.
) This quantity lies in the interval I ∈ [0, n ], where n is the average number of excitations in the system.The minimal value I min = 0 is obtained for classical states like Gaussian or thermal states, while purely quantum states, such as for example Fock and cat states, yield the maximum value of I max = n .Fig.4 depicts I and I max = n , with red triangles and blue dots respectively, for |Ψ(N ) = V (3) m (N ) |0 .As one can see, quantumness and population both increase approximately exponentially in time and the former nearly saturates the bound I max imposed by the latter.This witnesses the rapid evolution towards states of macroscopic character as well as their close-to-maximal non-classicality.

FIG. 4 .
FIG. 4. Comparative plot of the quantum estimator I (red triangles) and the average number of mechanical excitations nm (blue dots) as functions of the number of driving periods.The experimental parameters are set as η = 20, k = 1/60.

4 FIG. 5 .
FIG. 5. Fidelity between the state of the mirror computed via a third and a fourth order Magnus expansion as a function of the integration time, expressed in terms of mechanical driving periods.The experimental parameters are set as η = 20, k = 1/60.The graph indicates that high fidelity is obtained up to 20 driving periods: F3,4 0.985.

FIG. 7 .
FIG. 7. Comparison of the cut profiles p = 0 of the Wigner function of the state of the mirror after an evolution lasting 20 mechanical periods with the mirror initially in its ground state (blue dotted line) and two thermal states with respectively n th m ∼ 1 (green dashed line) and n th m ∼ 10 (red line).The experimental parameters are set as η = 20, k = 1/60 and ω = 2π × 10 7 Hz.

FIG. 8 .
Fidelity between the final state of the system (cavity plus mirror) in case of a mechanical damped evolution and the ideal scenario as a function of the mechanical quality factor Q = ω/γm.The total evolution is supposed to last 20 mechanical periods with the mirror initially in its ground state and the dimensionless driving and coupling respectively set as η = 20 and k = 1/60. 2π

FIG. 9 .
FIG. 9. Cavity occupation renormalized with respect to the population of the mirror nc = nc / nm as a function of the order of the decomposition of the step function d.In the top-right corner we plot an enlargement of the driving profiles defined by ϕ
Fidelity between the final state of the system (cavity plus mirror) in case of photon losses and the ideal scenario as a function of the cavity decay rate κ for an evolution lasting 20 mechanical periods.